What are the two independent ratios used to specify a line's parameters?

In summary: If we chose a specific "a" and "b", then c= -ax- by and c can be anything. On the other hand, if we want c= 0, then ax+ by= 0 and we can choose "a" and "b" so that is true. In summary, "degrees of freedom" for lines refers to the fact that a line has two degrees of freedom, meaning that two of the three parameters (a, b, and c) can be chosen freely while the third is then determined. This is represented by the notation {a : b : c} which stands for the proportion a/b = b/c. This notation allows for the inclusion of vertical lines, which is not possible
  • #1
sh86
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"Degrees of freedom" for lines

I'm reading something about "degrees of freedom" trying to learn what exactly it means, and there's this one sentence I'm running into that I can't really understand...
A line is specified by two parameters (the two independent ratios [tex]\{a : b : c\}[/tex]) and so has two degrees of freedom.

What is this "the two independent ratios {a : b : c}" ?

They talk a lot about how a line on a plane is represented by the equation [tex]ax+by+c=0[/tex]. But I know from learning about [tex]y=mx+b[/tex] in grade school that you only need two numbers to specify a line.. :confused: If anybody could explain that sentence to me I'd really appreciate it.
 
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  • #2


sh86 said:
I'm reading something about "degrees of freedom" trying to learn what exactly it means, and there's this one sentence I'm running into that I can't really understand...


What is this "the two independent ratios {a : b : c}" ?

They talk a lot about how a line on a plane is represented by the equation [tex]ax+by+c=0[/tex]. But I know from learning about [tex]y=mx+b[/tex] in grade school that you only need two numbers to specify a line.. :confused: If anybody could explain that sentence to me I'd really appreciate it.
The point that the author was trying to make is that to include ALL lines, you need to allow vertical lines (x=k). The form being used in the text allows for this (b=0). The two degrees of freedom is a way of saying that multiplying a,b,c by a constant doesn't change the line.
 
  • #3


{a:b:c} is shorthand for the proportion a/b= b/c. There are "two degrees of freedom" because you are "free" to choose two of the numbers to be almost anything you like and then could solve for the third.
 
  • #4


HallsofIvy said:
{a:b:c} is shorthand for the proportion a/b= b/c. There are "two degrees of freedom" because you are "free" to choose two of the numbers to be almost anything you like and then could solve for the third.

Wow, that {a:b:c} notation is confusing; I've never seen that.
 
  • #5


flatmaster said:
Wow, that {a:b:c} notation is confusing; I've never seen that.

The notation is not new to me, but the concept of two degrees of freedom for a straight line is new (to me). I always thought a caterpillar walking along a wire had only one degree of freedom, same as all straight lines regardless of where they are. How does introducing more constants into the equation change that? Halls, can you expand a bit on your explanation?
 
  • #6


Well, you are not a caterpillar, are you? If you were constrained to a specific straight line, but could pick any point on that line, yes, that would be "one degree of freedom". Here, however, If we write a line as "ax+ by+ c= 0", we could multiply or divide each of the coefficients by any number (except 0 of course) and still have the same line: "rax+ rby+ rc= 0" is satisfied by exactly the same (x,y) and so is the same line. Notice that ra/rb= a/b and rb/rc= b/c no matter what r is. In the formula "ax+ by+ c= 0" two of the numbers can be chosen any way we want but the other is then fixed.
 

What are degrees of freedom for lines?

Degrees of freedom for lines refers to the number of independent variables that can vary in a linear regression model. In other words, it is the number of data points minus the number of parameters estimated in the model.

Why are degrees of freedom important in linear regression?

Degrees of freedom are important in linear regression because they determine the amount of information available to estimate the parameters of the model. The more degrees of freedom, the more precise the estimates will be. Additionally, degrees of freedom are used in hypothesis testing and calculating the statistical significance of the model.

How can degrees of freedom be calculated for lines?

Degrees of freedom for lines can be calculated by subtracting the number of parameters estimated in the model from the total number of data points. For example, if a linear regression has 100 data points and 2 parameters (intercept and slope), the degrees of freedom would be 98.

What is the relationship between degrees of freedom and sample size?

There is a direct relationship between degrees of freedom and sample size. As the sample size increases, so does the degrees of freedom. This means that larger sample sizes provide more information and result in more precise estimates in a linear regression model.

Are there any limitations to using degrees of freedom in linear regression?

Yes, there are some limitations to using degrees of freedom in linear regression. One limitation is that it assumes that the data follows a normal distribution. Additionally, degrees of freedom only apply to linear regression models and may not be applicable in other types of statistical analyses.

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